[...]... view of undergraduate mathematics by approaching the subject through its history Since readers should have had some mathematical experience, certain basics are assumed and the mathematics is not developed formally as in a standard text On the other hand, the mathematics is pursued more thoroughly than in most general histories of mathematics, because mathematics is our main goal and history only the... to a deeper understanding of complex analysis and algebraic geometry, among other things Thus I hope that the book will be not only a xi xii Preface to the First Edition bird’s-eye view of undergraduate mathematics but also a glimpse of wider horizons Some historians of mathematics may object to my anachronistic use of modern notation and (fairly) modern interpretations of classical mathematics This... due to my failure to follow all their advice To them, and to Anne-Marie Vandenberg for her usual excellent typing, I offer my sincere thanks John Stillwell Monash University Victoria, Australia 1989 Contents Preface to the Third Edition vii Preface to the Second Edition ix Preface to the First Edition xi 1 2 3 The Theorem of Pythagoras 1.1 Arithmetic and Geometry 1.2 Pythagorean Triples 1.3... Rings and Fields 21.8 Biographical Notes: Dedekind, Hilbert, and Noether 439 440 442 445 448 452 454 457 459 22 Topology 22.1 Geometry and Topology 22.2 Polyhedron Formulas of Descartes and Euler 22.3 The Classification of Surfaces 22.4 Descartes and Gauss–Bonnet 22.5 Euler Characteristic and Curvature 22.6 Surfaces and Planes... describing points by numbers—their coordinates and describing each curve by an equation satisfied by the coordinates of its points This fusion of numbers with geometry is briefly explored at the end of this chapter, where we use the formula a2 + b2 = c2 to define the concept of distance in terms of coordinates J Stillwell, Mathematics and Its History, Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4419-6053-5... friends, colleagues, and reviewers who drew my attention to faults in the first edition, and helped me in the process of revision Special thanks go to the following people • My sons, Michael and Robert, who did most of the typing, and my wife, Elaine, who did a great deal of the proofreading • My students in Math 310 at the University of San Francisco, who tried out many of the exercises, and to Tristan Needham,... only the means of approaching it Readers are assumed to know basic calculus, algebra, and geometry, to understand the language of set theory, and to have met some more advanced topics such as group theory, topology, and differential equations I have tried to pick out the dominant themes of this body of mathematics, and to weave them together as strongly as possible by tracing their historical development... DeTemple, Wes Hughes, Christine Muldoon, Martin Muldoon, and Abe Shenitzer, for corrections and suggestions John Stillwell Monash University Victoria, Australia 2001 Preface to the First Edition One of the disappointments experienced by most mathematics students is that they never get a course on mathematics They get courses in calculus, algebra, topology, and so on, but the division of labor in teaching seems... known to have possessed the Pythagorean theorem; van der Waerden (1983) gives examples from China (between 200 bce and 220 ce) and India (between 500 and 200 bce) The most complete understanding of the problem in ancient times was achieved in Greek mathematics, between Euclid (around 300 bce) and Diophantus (around 250 ce) We now know that the general formula for generating Pythagorean triples is a = (p2... Contents xxi 23.8 Biographical Notes: Lie, Killing, and Cartan 518 24 Sets, Logic, and Computation 24.1 Sets 24.2 Ordinals 24.3 Measure 24.4 Axiom of Choice and Large Cardinals 24.5 The Diagonal Argument 24.6 Computability 24.7 Logic and G¨ del’s Theorem o 24.8 Provability and Truth 24.9 Biographical Notes: G¨ del . to http://www.springer.com/series/666 John Stillwell Mathematics and Its History Third Edition 123 John Stillwell Department of Mathematics University of San Francisco San. Michael, and Robert Preface to the Third Edition The aim of this book, announced in the first edition, is to give a bird’s- eye view of undergraduate mathematics